Comparison between two methods for pulse sequence

In order to test the performances of the proposed calibration procedure, in terms of reproduction of the main statistical features of the recorded streamflow series, the generation of synthetic flow series was required. For this purpose, the generation of synthetic effective rainfall series was preliminary necessary (through the universal random variable generator described in Section 3), to subsequently allow the discrete-time convolution of these series with the time response function of the basin. An even more necessary operation, however, consisted in the estimation of the parameters of the so-called PWNE model of effective rainfall, whose PDF and CDF are given, respectively, by Eq. (3.29) and (3.30). In Section 3 some details have been provided on the available techniques for the assessment of the sequence of rainfall pulses, starting from the recorded streamflow series, from which the parameters of the PWNE model thus estimated. In particular, the author focused on: the FPOT procedure, proposed by Claps and Laio [2003], and the procedure proposed by Xu et al. [2001], to which, from this moment, the author will refer as MA (Minimization Approach).

Starting from this general framework, in the present subsection the intention of the author is to provide the reader with a comparison between results obtained by the adoption of the FPOT and the MA procedures.

The first tested methodology is the FPOT procedure. As already mentioned in Section 3, the author highlights that, once estimated the sequence of

Section 4. Application and testing

90 Filtered Peaks (FP, obtained by subtracting to each local maximum, of the streamflow series, the first previous local minimum), a threshold filter has to be increasingly applied to the FP series, until the independence test of Kendall and the Poisson distribution test of Cunnane are jointly met [Claps and Laio, 2003]. The author will refer to the peaks that meet these tests as significant FPOT peaks, which are graphically reported, for the first recording year of the streamflow process, in Fig. 4.5, for each case study series.

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91

Fig. 4.5. Significant FPOT peaks in the first year of recording – (a) Alento; (b)

Section 4. Application and testing

92 The MA procedure by Xu et al. [2001] was also applied to each recorded streamflow series (for more details about the methodology, the interested reader may refer to Subsection 3.5.3.2). In particular, it is worth underlying that the pulse heights obtained by the application of MA procedure (from now called significant MA peaks) were estimated through the minimization of the cost function shown by Eq. (3.32). Given that, according to Xu et al. [2001], pulses occur on all those days when streamflow increases, it is obvious that the minimization of Eq. (3.32) defines a highly multidimensional minimization problem that the author solved by the adoption of the Powell‟s method described in Subsection 4.3.1. The estimated peak sequences, for each case study and for the first year of recording, are graphically reported in Fig. 4.6.

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93

Fig. 4.6. Significant MA peaks in the first year of recording – (a) Alento; (b)

Section 4. Application and testing

94 Once estimated the significant peak sequences, the author verified that the storm events occurred along the time axis according to a non-homogeneous Poisson point process, as required by Xu et al. [2001]. For this purpose, a goodness-of-fit test was undertaken for each month of the year (Tab. 4.4 provides values for each of the 3 case study watersheds).

Tab. 4.5. goodness-of-fit test for non-homogeneous Poisson distribution on significant MA peaks Series (a) (b) (c) Month χ2 value January 1.335 0.318 0.923 February 0.556 1.072 0.079 March 0.013 0.747 0.616 April 0.080 0.344 0.989 May 0.279 1.802 2.486 June 1.083 0.490 3.023 July 0.538 0.008 0.436 August 0.003 0.670 0.372 September 1.309 2.472 0.088 October 0.025 0.041 0.002 November 0.024 1.469 0.148 December 0.151 0.508 0.001

Given a statistic limit of 9.488 at the 5% significance level (see Xu et al. [2001]), it is evident that for each basin the null hypothesis of Poisson distributed occurrences during each month cannot be rejected. The non- homogeneous Poisson point process is therefore acceptable for describing the storm occurrences. Nevertheless, as above introduced, the slight differences in the mean number of rainfall events between months allow the author to assume the input process to be stationary.

Section 4. Application and testing

95 One major comment is needed about the comparison of the two effective peaks estimation procedures. As it is possible to realize also by a graphical analysis of Fig. 4.5 and 4.6, the mean number of significant rainfall events per year obtained by the MA procedure is considerably higher than that resulting from the FPOT procedure, for each of the case studies. The values reported in Tab. 4.5 confirm this consideration.

Tab. 4.6. Mean number of rainfall events per year: comparison between MA and

FPOT procedures

Series (a) (b) (c)

Procedures Mean annual rainfall events

FPOT 17.5 13.1 16.0

MA 39.4 17.9 46.7

In particular, the mean number of events per year obtained by the FPOT approach confirms the results of Claps and Laio [2005], who reported a rough estimate of about 5-20 peaks per year for each of the numerous analyzed runoff series. However, the main point consists in the fact that authors themselves admitted that such estimation could result in a underestimation of the actual number of effective rainfall events, suggesting a clue of the inadequacy of the Poisson independent model in the correct reproduction of the effective rainfall behavior. The mean number of events per year obtained by the MA procedure, instead, seems to be much more compatible with the actual number of events. As a matter of fact, taking the Alento River as an example, several regional studies undertaken on the

Section 4. Application and testing

96 Italian territory on several uniform compartments (from a hydrological point of view) seem to confirm the result. In particular, using the Extreme Value-I (EV-I) distribution for statistical analysis of rainfall heights‟ annual maxima in intervals of different duration (in the area where the Alento basin is located), a mean number of events per year equal to 37 was evaluated [Di Nunno, 1981]. Furthermore, adopting the Two-Component Extreme Value (TCEV) distribution [Rossi et al., 1984] for the analysis of instantaneous discharges annual maxima at Casalvelino station (Alento River), a mean number of flood events per year equal to 44 was estimated [Rossi and Villani, 1995].

Starting from these considerations, the author considered the MA procedure more efficient, compared to the FPOT approach, thanks to its ability to provide a more realistic pulses sequence in terms of mean annual number of events. It is worth underlying that, on the contrary, a comparison between FPOT and MA procedures in terms of the mean annual intensity of rainfall events would be meaningless. Indeed, while the FPOT approach derives the peak sequence only by means of operation on the recorded runoff series, the MA procedure provides a sequence, through minimization of Eq. (3.32), strictly dependent on the values of the estimated parameters of the TF. Furthermore, given that TFs estimates obtained in this work are considerably different from those provided by Murrone et al. [1997], concerning the Alento River, and Claps and Laio [2005], concerning the Scrivia and Bormida rivers, a comparison between pulses sequence in terms of intensity would be practically useless.

Eventually, a first way to ascertain, on the one hand, the validity of the response function form adopted in the calibration approach, and, on the other hand, the efficiency of the selected MA procedure for the determination of the effective peaks series, is to provide the reader with a graphical

Section 4. Application and testing

97 comparison between recorded and reconstructed series (Fig. 4.7). The latter one, in particular, was obtained, for each case study series, through the convolution (see Eq. 3.34) between the known response function and the corresponding input sequence. From a quick graphical analysis of the proposed comparison, it is possible to observe the almost perfect superimposition of the peak occurrences time between reconstructed and recorded series, in addition to a practical coincidence between corresponding discharge values.

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98

Fig. 4.7. Recorded and reconstructed runoff series comparison – (a) Alento; (b)

Section 4. Application and testing

99